Evaluating a Function of Two Variables

Let \(f(x,y)=\sqrt{x}+x\sqrt{y}\). Find:

1. \(f(1,4)\)
2. \(\!f(a^{2},9b^{2})\), \(a>0\), \(b>0\)
3. \(f(x+\Delta x, y)\)
4. \(f(x,y+\Delta y)\)

Solution (a) \(f(1,4)=\sqrt{1}+1\sqrt{4}=1+2=3\qquad\hspace{5.2pt}\) \(\color{#0066A7}{x=1; y=4}\).

(b)\(f(a^{2},9b^{2})=\sqrt{a^{2}}+a^{2} \sqrt{9b^{2}}=a+3a^{2}b\quad \color{#0066A7}{x=a^{2}; y=9b^{2}; a>0; b>0.}\)

(c) \(f(x+\Delta x, y)=\sqrt{x+\Delta x}+( x+\Delta x) \sqrt{y}\)

(d) \(f(x,y+\Delta y)=\sqrt{x}+x\sqrt{y+\Delta y}\)